Adiabatic Radiofrequency
Pulse Forms in Biomedical
Nuclear Magnetic
Resonance
DAVID G. NORRIS∗
Max-Planck-Institute of Cognitive Neuroscience, Stephanstr. 1a, D-04103 Leipzig, Germany
ABSTRACT: Adiabatic radio frequency (RF) pulses are in widespread use in biomedical
magnetic resonance imaging and spectroscopy. The primary advantage of adiabatic pulses is
that provided the condition for adiabaticity is satisfied they can be made insensitive to inhomogeneities in the RF field. In this pedagogical article the principles of adiabatic fast passage
(AFP) are explained, and the use of AFP to invert both stationary and flowing spin systems
is examined. The hyperbolic secant pulse is presented as a pulse capable of performing slice
selective adiabatic inversion. Lower power alternatives to this pulse are described, and the principle of offset independent constant adiabaticity is elucidated. Instantaneous reversal of the
orientation of the effective RF field as a means of producing excitation and refocusing pulses
is presented, as are methods of obtaining slice selective excitation with these pulses. © 2002
Wiley Periodicals, Inc.
KEY WORDS:
Concepts Magn Reson 14: 89–101 2002
adiabatic RF; adiabatic fast passage; hyperbolic secant pulse
INTRODUCTION
Radio frequency (RF) pulse forms that rely on the
principle of adiabatic fast passage (AFP) (1) have
found widespread utility in biomedical nuclear
magnetic resonance (NMR). Pulses based on AFP
cannot at present be designed using the formalism
developed by Shinnar and Le Roux (2–8) and are
Received 18 June 2001; revised 23 October 2001;
accepted 2 November 2001.
∗
Present address: FC Donders Centre for Cognitive Neuroimaging, Trigon 181, P.O. Box 9101, NL 6500 HB Nijmegen,
The Netherlands; e-mail: [email protected].
Concepts in Magnetic Resonance, Vol. 14(2) 89–101 (2002)
DOI 10.1002/cmr.10007
© 2002 Wiley Periodicals, Inc.
best understood by recourse to a mixture of vector
sweep-diagrams and mathematical arguments. The
attraction of AFP pulses is their insensitivity, over
a broad range, to inhomogeneities in the B1 -field.
This is particularly important in biomedical applications because of the desire to examine relatively
large volumes, up to the size of the human torso,
or in some situations to employ surface coils both
for RF transmission and signal reception.
The purpose of this article is educational and
is aimed at acquainting the reader with the principles of adiabatic fast passage, and the adiabatic
RF pulses and some of the applications which
are encountered in biomedical NMR. It is not
intended to provide an all encompassing review.
For a review of the use of adiabatic pulses in
biomedical NMR the reader is referred to (9).
89
90
NORRIS
ADIABATIC FAST PASSAGE
The classical understanding of AFP is based on
a precession argument in the rotating frame of
reference. It is well known that in the frame rotating at the Larmor frequency 0 the magnetisation
vector M will precess about the effective B1 -field
B1e . For a pulse having a frequency which differs
from 0 by this effective field is given by
B1e = B1 +
k
[1]
where B1 is a vector in the transverse plane of the
rotating coordinate system and k is a unit vector
along the z -axis. In this article trajectories of B1e
will generally be considered in a frame of reference rotating with the instantaneous frequency of
the RF pulse. In this frame sweeps are most easily
visualized; occasionally a second frame of reference will be used, denoted by x y z in which
the z -axis rotates in parallel with the B1e -field.
will often be refFor convenience the quantity fered to as Bz . The orientation of these vectors
is indicated in Fig. 1. In a standard RF pulse
the frequency is held constant, and the precession
about B1e is used to change the orientation of M,
so of necessity the angle between B1e and M is
large. In AFP the orientation of B1e is changed
B1e, z''
z'
M
Bz
y'
θ
Ω
B1
x'
Figure 1 Sketch showing the effective magnetic field
B1e in the frame of reference rotating with the instantaneous frequency of the RF pulse, and its two component vectors B1 and Bz , where Bz = . The angle
is defined as that subtended by B1e and B1 . The
magnetization M is shown as a shaded cone precessing about B1e . The adiabatic sweep is described by a
rotation about the vector which is shown as lying in
the x y -plane. The z -axis of the frame of reference
that rotates with B1e is also shown. The x - and y -axes
would initially be aligned with the x - and y -axes and
rotate with B1e but have been omitted for clarity.
in such a way that the angle subtended by M and
B1e remains constant throughout the pulse, and M
hence moves with B1e . The term “adiabatic” arises
because the change in the orientation of B1e must
be sufficiently slow that M is able to follow it.
However, the total duration of the pulse must be
shorter than any relaxation processes and hence
the term “fast passage.” Following Abragam (1)
we can consider the motion of M in the reference frame x y z . The motion of B1e can be
described by a rotation about a vector perpendicular to the z -axis as is shown in Fig. 1. The
equation of motion of M in this reference frame
is given by the well known equation for transformation to a rotating coordinate system by
M
= M × B1e +
[2]
t
If
|
| |B1e |
[3]
and remembering that in this frame B1e is aligned
parallel to the z -axis, then we can write
Mx y
∼
= |B1e |Mx y t
[4]
The solution of Eq. [4] represents a rotation at
constant magnitude, and hence a constant angle
of precession of M about B1e . Equation [3] represents the most fundamental condition for adiabaticity. If we now include the consideration that
an adiabatic pulse must be faster than any relaxation processes then we have
1 1
|
| |B1e |
T2 T2∗
[5]
Classically only T2 relaxation is considered, which
sets an absolute physical limit for the pulse
duration which is independent of experimental
conditions. However, particularly for in vivo applications, T2∗ may be more critical.
If we consider a rotation of B1e from the positive to the negative z -axis, i.e., an adiabatic
inversion, in which the amplitude of the B1 field is
held constant and the sweep is effected solely by
modulation of the frequency offset then
tan =
|Bz |
|B1 |
[6]
and as only Bz varies with time
˙ sec2 = −
|
|B1 |
Ḃz |
[7]
RF PULSE FORMS IN BIOMEDICAL NMR
After some manipulation it is then possible to
write.
= ˙ = −
B1 Ḃz
B21e
[8]
Insertion of the inequality given in Eq. [3] gives
the result that
3 B |Ḃz | 1e [9]
B1
This condition is most sorely tested when B1e
passes through the transverse plane and has its
minimum value of B1e = B1 at which point Eq. [9]
simplifies to
|Ḃz | B21 [10]
On resonance the trajectory of B1e will pass
through the transverse plane midway throughout
the sweep. In the presence of a frequency offset
the passage through the transverse plane will be
shifted in time dependent upon the size of the
frequency offset relative to the frequency width
of the sweep. The inversion will still be successful provided that this frequency width is sufficiently large that B1e is aligned along the z -axis
both at the start and at the end of the sweep.
Provided that Eq. [10] is satisfied, inversion will
occur over a range of B1 values. The efficiency
of the inversion will also be independent of the
exact value of B1 within this range. These properties made AFP attractive for inversion in the presence of significant inhomogeneities in both the B0
and the B1 fields (10). Termination of the sweep
at the halfway point will result in a saturation
pulse which will be similarly insensitive to inhomogeneities in B1 as the inversion pulse. However,
the efficiency of the saturation will depend on the
frequency offset and hence saturation pulses will
show some sensitivity to inhomogeneity in B0 , as
will be discussed below in the context of halfsech pulses. It is an important general point that
this sensitivity to off-resonance effects makes simple AFP sweeps generally unsuitable for excitation
and refocusing, and for use in combination with
slice selection employing magnetic field gradients.
AFP may be achieved by three methods in practice: by holding the frequency of the RF-pulse
constant and sweeping the B0 -field, by sweeping
the frequency of the RF-pulse, and by appropriately modulating the phase of the RF-pulse.
Historically the first of these variants was of
importance in the days of CW spectrometers as
91
the RF was held constant and the magnetic field
swept through resonance. In modern systems it is
generally more convenient to modulate the phase
than to directly vary the frequency: most consoles
only permit the transmission of RF pulses at a
fixed carrier frequency, but the phase can generally be freely programmed at the generation of the
pulse.
An interesting extension of the application of
AFP is for the inversion of flowing spins. A
magnetic field gradient is applied parallel to the
direction of the flow as shown in Fig. 2. This
experiment is really an extension of the AFP
implementation in which the RF is held constant
and the B0 -field is swept. Motion in the direction
of the gradient is used to modulate the instantaneous Larmor frequency, so spins moving with a
constant velocity will experience a linear sweep of
the resonance frequency, as shown in Fig. 2. Typically a small transmitter coil is used to effect the
inversion, for example, placed above the carotid
artery in the neck. For maximum efficiency the
frequency of the RF is set equal to the Larmor
frequency at the center of the coil in the presence of the gradient. As in the inversion experiment described above, the B1 -field is constant and
so the adiabatic condition is most tested at zero
offset frequency. For spins moving with a velocity
component v parallel to the gradient field we have
= Gvt
[11]
where for convenience the temporal origin is
taken as the time at which there is zero frequency
Figure 2 Flow induced adiabatic inversion. The spins
flowing with velocity v experience a linear variation in
the frequency offset due to the presence of the magnetic
field gradient. The RF pulse frequency should be so
adjusted that the midpoint of the inversion is roughly at
the center of the surface coil.
92
NORRIS
offset. In the general situation
= ˙
[12]
where
tan =
Gvt
|B1 |
[13]
for small values of tan ∼
= and hence Eq. [3]
can be rewritten as
Gv [14]
B |B1 |
1
The inversion must still be faster than the appropriate relaxation time for the blood as described in
Eq. [5], but there is also the additional condition
that the inversion must take place while the spins
are within the sensitive region of the labelling coil.
This leads to the further condition that
Gv v
1
|B1 |
[15]
T
d
B 2
1
where d is the diameter of the coil. Contravention
of the adiabatic condition can hence occur in
three situations: for slow moving spins if relaxation
occurs during the transition, i.e., if the left most
inequality of [15] is contravened; if the choice
of gradient and B1 -fields is inappropriate for the
diameter of the coil, corresponding to the central
inequality of Eq. [15]; and finally if the velocity is
so great that the fundemental adiabatic condition
is contravened which is the right hand inequality
in Eq. [15].
Velocity-dependent adiabatic inversion was initially proposed for performing projective angiography in the brain (11) and was subsequently
proposed as a method for performing continuous
arterial spin-labelling (12–15). Two studies have
also examined the efficiency of the labelling process using this technique 16 17.
The approaches described above were developed during the early days of NMR. With the
advent of imaging systems and the requirement
for slice selective excitation it became possible to
modulate the form of the B1 -field. The potential advantage of such a sweep is the consideration that if a linear sweep satisfies the adiabatic
condition at resonance then this condition will
be exceeded by more than required for all other
offsets, leading to longer pulses and/or higher RF
power than necessary. One obvious modulation
form is the sin/cos form (18)
B1 t ∝ sin t
[16]
∝ cos t
[17]
which ensures that B1e is constant throughout the
pulse as of course is . This modulation form
ensures that the adiabaticity is constant throughout the pulse, but no attempt is made to minimize
the pulse duration. A natural attempt to do this is
to hold both the adiabaticity and the B1 -field constant and then to vary the offset frequency. This
gives rise to the constant adiabaticity pulse form
19 20, which may be derived as follows: The
inequality given in Eq. [3] may be converted into
an equation by introducing the adiabaticity factor
K, such that
|K
| = |B1e |
[18]
By reference to Fig. 1 we know that
B1e =
B1
cos [19]
and so it is possible to write Eq. [18] as
B |˙ cos | = 1 K
[20]
which, if we take the temporal origin t = 0 to
be at the center of the sweep, may be integrated
to give
B t | sin | = 1 [21]
K
but
= B1 tan [22]
and we can hence write
∝ 2 B21 t
K 2 − B1 t2 [23]
This form has the advantage that the minimum
pulse duration should always be achieved, but at
the start and end of the sweep effectively an infinite frequency offset is required to ensure that the
B1e -field is parallel or antiparallel to the z -axis.
It is hence clear that superior performance should
be expected from pulse shapes that have a tapered
amplitude modulation at the start and end of the
sweep. Other commonly encountered modulation
forms are the constant/tan form (21), and the
sech/tanh form which will be examined in detail in
the next section. Some attempts have been made
to optimize the performance of these sweep functions so that they can be guaranteed to function
over a specified range of variation in the B1 -field,
using numerical approaches (22) and for some
cases analytical solutions (23).
RF PULSE FORMS IN BIOMEDICAL NMR
SLICE SELECTIVE INVERSION
AND SATURATION
discussed in (26) are:
A natural limitation of the AFP forms discussed
in the previous section is that these waveforms do
not have an off-resonance behavior which makes
them suitable for slice selective excitation. The
first pulse to be found that produces a reasonably
sharp response profile for inversion was the hyperbolic secant pulse (19 24–26) and its variants,
which will be discussed in this section.
THE HYPERBOLIC SECANT PULSE
The hyperbolic secant or sech pulse has the complex form
1 t = B0 secht1+i
B
1
[24]
or
1 t = B0 secht expi ln sech t
B
1
[25]
which represents both an amplitude and a frequency modulation and may be explicitly written
as
B1 t = B10 secht
t = − tanht
[26]
The corresponding amplitude and phase modulation is illustrated in Fig. 3. The parameter determines the degree of phase modulation and
in combination with the parameter , which gives
the truncation level, determines the bandwidth
according to
BW = 2
93
[27]
It is possible to analytically solve the Bloch equations using the time-dependent B1 -field given in
Eq. [24] as a driving function. The result is a
broadband inversion with a fairly sharp inversion
profile as shown in Fig. 4 for a range of values. Above a certain threshold in the B1 -field
strength the inversion profile is largely independent of this parameter. The analytical solution of
the Bloch equation is complex (26) and shall not
be given here. There are some specific insights
to be gained from the analysis, one being the
derivation of Eq. [27]. Further salient features as
1. In the limit of no phase modulation = 0,
when
B01 ≈ n
[28]
and n is an integer, the magnetization
remains unaltered by the pulse. This is similar to the phenomenon of optical transparency and means that for the correct
value of the B1 -field strength a 2n rotation
is achieved, independent of the frequency
offset.
2. In the limit in which →∞, and →0 =
constant, an adiabatic fast passage is effectively performed.
3. Provided that ≥ 2 and B01 ≥ the
inversion is essentially independent of the
strength of B01 , a result given above.
Taken together these results show that the pulse
has an adiabatic quality but also that the nature
of the amplitude modulation is significant. Intuitively it is possible to understand the performance
of the sech pulse in terms of the similarities to an
adiabatic passage: as such, useful insights into the
performance of the sech pulse have been gained
by using sweep diagrams (27). The sweep diagram
makes it possible to understand the insensitivity to
the B1 -field strength, as shown in Fig. 5(A). The
start and end points of the trajectory are unaffected by variations in the B1 -field strength; only
the locus of the sweep is modified. By considering
a constant offset frequency it is also possible to
understand the slice selective nature of the pulse:
within the slice the frequency offset is smaller than
the maximum frequency offset of the pulse and
so the adiabatic passage still results in a reversal of the magnetization vector. Outside the slice
the frequency offset is larger and so the orientation of B1e cannot be reversed. These processes
are illustrated in Fig. 5(B). It should be noted
that outside the selected slice the magnetization
is tipped out of the longitudinal axis, and subsequently returned to it. If the adiabatic condition is
contravened by the pulse duration being comparable to T2 then the longitudinal magnetization outside the selected slice will be attenuated as a result
of its excursion into the transverse plane with deleterious effects on the slice profile 28 29.
It is also possible to consider using half of a
sech pulse in order to perform an adiabatic saturation (27). Given the sweep trajectories discussed
94
NORRIS
100
Pulse Amplitude (%)
80
60
40
20
0
0
200
400
600
Duration (ms)
800
1000
(a)
Phase (degrees)
180
0
180
0
200
400
600
Duration (ms)
800
1000
(b)
Figure 3
Amplitude (a) and phase modulation (b) for a sech pulse with = 5.
above it is hardly surprising that the excitation
profile is radically different than that of the full
sech pulse: the half-sech will tip the magnetization
away from the longitudinal axis, without returning
it there in the second half of the pulse, resulting in
a far broader excitation profile. At the frequency
offset = the initial frequency offset of the
pulse will be exactly cancelled by the external offset (cf. Eq. [26]) and as the initial amplitude of the
sech is also zero the adiabatic condition cannot
be satisfied at this offset 27 30. This results in
a notch of the excitation profile, and the phase of
the transverse magnetization is reversed in going
across this singularity because on one side of the
notch the magnitude of the initial frequency offset of the pulse will be greater than the external
offset, causing B1e to be antiparallel to the initial
direction of M0 whereas on the other side of the
RF PULSE FORMS IN BIOMEDICAL NMR
95
1
0.5
(a)
(b)
(c)
0
0.5
1
4000
2000
0
2000
4000
Figure 4 Inversion profiles obtained for a 10 ms sech pulse having the -values of 2 (a),
5 (b), and 10 (c). The increase in both bandwidth and profile quality with increasing -value
is clearly visible.
notch the two vectors are parallel as is shown in
Fig. 6 (27). The final orientation of the magnetization relative to the transverse plane is dependent only on the external frequency offset and the
strength of the B1 -field. This leads to the interesting result that the bandwidth is dependent only
on the strength of the B1 -field and not on the
z'
(a)
(b)
z'
pulse duration, provided of course that the adiabatic condition is still satisfied 27 30.
EXTENSIONS TO THE HYPERBOLIC
SECANT PULSE
Although the hyperbolic secant pulse was very successful at producing broadband inversion, it does
suffer from a relatively high peak RF power and in
B1e
B1e
(a)
B1
x'
x'
z'
(b)
B1
z'
B1e
x'
M
B1
x'
M
Figure 5 Sweep diagrams for slice selective adiabatic
inversion. (a) shows the sweep for two different values
of B1 -field strength and makes the point that although
the trajectories may differ significantly the same endpoint will be reached. (b) Shows how magnetization just
outside the selected slice is tipped away from the z -axis,
only to be returned to it at the end of the pulse. This is
because the maximum frequency offset of the pulse is
smaller than that due to the selection gradient.
Figure 6 Illustration of the phase reversal in the magnetization (M) that occurs at the frequency offset .
In (a) the magnitude of the initial frequency offset of
the pulse, , is greater than and the trajectory
of B1e starts from the −z -axis. In (b) the converse is
true. Although the final orientation of the B1e vectors
is in both instances similar, the differing initial polarity
causes the phase reversal in M.
96
NORRIS
some in vivo applications, such as localized spectroscopy using the ISIS technique (31) and perfusion imaging using pulsed arterial spin labelling
methods, a sharper inversion profile could be
desired. On the basis of the sweep diagram
approach it is straightforward to conceive that any
trajectory that smoothly reverses its orientation
along the z -axis, by starting and finishing with a
large frequency offset and a low amplitude of the
B1 -field, will be capable of producing a slice selective inversion. Initial attempts to improve upon
the performance of the sech pulse concentrated
on producing pulses which satisfy the requirement
for a low initial and final amplitude of the B1 -field
with a constant B1 -field intensity during the central period of the pulse. Using considerations of
this nature Rosenfeld et al. produced a pulse
made up of three types of segment which were
smoothly joined (32). The performance of this
pulse was subsequently optimized 33 34. Using
similar considerations Kupc̆e and Freeman (35)
developed the WURST pulse (an acronym ostensibly based on the formulation wideband uniform
rate and smooth truncation, but also invoked
because the amplitude form resembles that of a
sausage). This has the analytical form for the modulation of the B1 -field of
B1 t = B10 1 − | sint|n [29]
with the frequency offset being swept as a linear
function of time. The higher the value of n, the
sharper the transition in the B1 -field at the ends of
the pulse, and the higher the bandwidth. The value
n = 20 was recommended in the original paper on
this subject (35). In comparison to the sech pulse
the WURST-20 pulse has a broader bandwidth but
a somewhat shallower transition gradient. Shen
(36) used a previously published transform (19)
to rescale the temporal axis of known modulation
functions, thus achieving a significant reduction in
RF power, while maximizing the adiabaticity at
resonance.
However, the approach that gives the most
insight into the design of slice selective inversion
pulses is that developed by Kupc̆e and Freeman
37 38 and Tannús and Garwood (39). This is
an extension of the constant adiabaticity sweep
described above in Eqs. [18]–[23]. The salient
point in this approach is the realization that the
degree of adiabaticity across the whole bandwidth
of the pulse is important and not just at resonance. If we allow both the frequency offset and
the amplitude of the B1 -field to be time-varying,
then we can rewrite Eq. [9] as
K=
B21 + B2z 3/2
Ḃz B1 − Ḃ1 Bz
[30]
For spins that are off-resonance by the frequency
Bz can be written as
Bz t =
− F t
[31]
where F t represents the modulation function
for Bz . For all frequency offsets within the selected
slice K should be independent of . For each within the slice there will be a time, t , at which
the resonance condition is satisfied and Bz t will be zero. However, K is also required to have
this value for all other offsets within the slice.
Equation [30] can then be rewritten as
Kt Ḃz = B21
[32]
This equation bears a remarkable similarity to
Eq. [10]. So once a suitable modulation form
for the amplitude has been determined then the
frequency modulation follows automatically from
Eq. [32]. A key feature of this approach is that
the power is distributed evenly over the whole of
the inversion bandwidth. An interesting observation is that a nonadiabatic pulse with a rectangular slice profile distributes the energy equally
over the bandwidth at any given time, whereas an
adiabatic pulse does this sequentially (39). There
is still an infinite number of pulses that satisfy
Eq. [32], including, notably the sech pulse itself,
but forms can be chosen which use low peak and
mean B1 -fields, or further optimization procedures
can be sought 40 41. A number of pulse forms
are given in Refs. 38 and 39; in particular the
WURST pulse in its original form does not satisfy Eq. [32], but a frequency modulation function
is given in table 1 of Ref. 39 that remedies this
deficit.
The realization that the inversion occurs
sequentially through the slice when using the sech
pulse leads to the idea that by increasing the magnetic field gradient at the start and end of the
pulse, sharper edges and a reduced chemical shift
artifact can be achieved relative to the original
sech pulse, but at no increase in RF power (42).
These pulses are collectively known as frequency
offset corrected inversion pulses. A number of
RF PULSE FORMS IN BIOMEDICAL NMR
possible trajectories were presented in the original publication on this topic (42) and were subsequently improved upon by incorporating the gradient field into the principle of offset independent
constant adiabaticity (9).
z'
(a)
(b)
B1e
97
z'
M0
x'
M0
y'
B1e
y'
x'
EXCITATION AND
REFOCUSING PULSES
As discussed above, single adiabatic sweeps are
incapable of performing excitation or refocusing. In this section the combination of adiabatic
sweeps to achieve this end will be examined. The
general principle is that by combining sweeps
with different parameters the desired manipulation can be achieved and undesirable effects eliminated. Pulses which have a sufficiently well defined
behavior off-resonance can be considered for slice
selective schemes.
Non Slice Selective Pulses
The basic principle by which excitation or refocusing can be achieved is by utilizing instantaneous reversal of the orientation of the effective
B1e -field 43 44. The principles of operation of a
90◦ (44) excitation pulse and a 180◦ pulse (43) are
shown in Fig. 7. Considering the case of the 90◦
pulse, known as a BIR-1 pulse, the B1e -field is initially in the transverse plane, for example, initially
aligned along the x -axis (cf. Fig. 7(A)), and is
rotated to the longitudinal axis. Provided that the
adiabatic condition is satisfied the magnetization
initially along the z -axis will remain in a plane
perpendicular to the B1e -field but will fan out due
to the effects of inhomogeneities in the B1 -field
(cf. Fig. 7(B)). Once the B1e -field is aligned with
the z -axis it undergoes instantaneous reversal to
the −z -axis (cf. Fig. 7(C)). The B1e -field orientation is then returned to the transverse plane. For
a 90◦ pulse it should be returned to being parallel or antiparallel to the y -axis (cf. Fig. 7(D)); for
a 180◦ pulse it should be returned to the x -axis
(cf. Fig. 7(E)). The effect of instantaneous reversal is to reverse the effect of B1 -inhomogeneities in
a similar fashion to the way in which the refocusing pulse in a spin–echo experiment reverses the
effect of B0 -inhomogeneities. Provided that the
duration of the two halves of the pulse is equal,
and the locus of the sweep of the B1e -field during the second half of the pulse has the same
form (but not necessarily the same phase) as during the first half of the pulse, then the effects of
z'
(c)
z'
(d)
M0
y'
x'
M0
x'
y'
B1e
B1e
z'
(e)
z'
(f)
M0
B1e
B1e
y'
x'
x'
y'
M0
Figure 7 The BIR-1 pulse scheme. (a) shows the initial position of the magnetization vector M0 and the
B1e -field. It the time period between (a) and (b) B1e is
swept from the x- to the z -axis, and the magnetization
dephases in the transverse plane due to the combined
effects of inhomgeneity in B0 and B1 . The orientation
of B1e is then reversed to the −z -axis as shown in (c).
For a 90◦ excitation the B1e -field is swept to the y -axis
as shown in (d), for a 180◦ pulse it is swept back to the
x -axis as shown in (e), and for an arbitrary pulse angle
it is swept back at an angle to the −x -axis (f).
B1 -inhomogeneities will be exactly compensated.
The pulse excitation angle is determined solely by
the angle subtended by the initial and final orientations of the B1e -field, and it is hence possible to
program arbitrary excitation angles, independent
of B1 -field strength (45), as shown in Fig. 7(F).
As a general remark, it should be noted that
instantaneous reversal of the B1e -field orientation
can take place along any arbitrary axis. For future
use it is valuable to define a concise notation
which describes the sweep patterns of this type of
pulse. The pulse shown in Fig. 7 can be described
98
NORRIS
as P −1 P0 , a convention that was first followed
in the design of composite pulses 46 47. P
denotes a rotation of 90◦ about an axis at an angle
in the transverse plane as shown in Fig. 7(F).
The inverse sign denotes a sweep from the transverse to the longitudinal axis.
The main weakness of the pulse schemes
shown in Fig. 7 is that off-resonance effects are
not compensated: off-resonance there will be a
constant component of the B1e -field along the
z -axis that will not be reversed midway through
the pulse. These pulses hence have an intrinsically narrow bandwidth. At the cost of increasing the pulse-duration it is possible to reduce
this effect if the resonance offset of the adiabatic pulse never changes sign: one such scheme is
P180− −1 P0 P180 −1 P0 , known as BIR-2, in which
the instantaneous reversal of the B1e -field takes
place in the transverse plane. However, even
with this pulse there is a deleterious effect offresonance, as the constant frequency offset will tip
the B1e -field away from the transverse plane, with
the result that after the reversal in the orientation of this field the plane in which the magnetization vectors will lie will no longer be perpendicular
to B1e .
A number of adiabatic pulse schemes of this
general nature have been proposed, and the most
important are listed in Table 1 (43 44 48–50).
Of these pulses, that which has gained the most
widespread acceptance is that known as BIR-4
which may, at the simplest level, be viewed as
a concatenation of two BIR-1 pulses. By doubling the length of the pulse it is possible to
compensate for the off-resonance effects to which
BIR-1 is so sensitive by ensuring that the effect
of the off-resonance effects is equal and opposite between the two halves of the pulse. The
total pulse angle may be arrived at by modifying the trajectory of the B1e -field after either or
Table 1
The Major B1 Insensitive Pulse Schemesa
Pulse Name
Reference
Scheme
−1
BIR-1
44
P P0
BIR-2
44
P180− −1 P0 P180 −1 P0
BIREF-1
48
P0 P180 −1
BIREF-2a
48
P180 −1 P0
BIR-4
49, 50
P180 −1 P180+ 2 P 2 −1 P0
a
The pulses described in Ref. 48 are exclusively refocusing
pulses; the BIR pulses can be used to produce any excitation
angle.
both of the phase reversals, in an analagous manner as for BIR-1 (45). There is, however, a particular advantage to doing this in a symmetrical
fashion (50), so that the pulse train is written as
P180 −1 P180+/2 P/2 −1 P0 , to give an excitation
angle of , as the pulse then belongs to a class of
pulses having reflection symmetry in time. Similar pulse trains have been examined in the context
of hard composite pulses that compensate for the
effects of RF inhomogeneity 46 47 where it was
found that only pulses with a reflection symmetry in time could combine an insensitivity to RF
inhomogeneity with a broad excitation bandwidth.
Although the bandwidth of BIR-4 is acceptably
broad, the edges of the profile are not sufficiently sharp to make it useable for slice selective
excitation.
Slice Selective Pulses
There have been several attempts to combine the
desirable properties of adiabatic pulses with slice
selection using magnetic field gradients. The first
of these is based on using the properties of the
sech pulse and trying to compensate the undesirable phase dispersion caused by off-resonance
effects. The first proposal (51) is to form a composite pulse consisting of three successive sech
pulses in which the duration of the first two pulses
is half that of the last. The first two pulses give a
rotation of 360◦ with a phase error that is exactly
opposite that generated by the last 180◦ pulse.
The compensation effect is achieved by instantaneously reversing the value of the initial frequency
offset between the second and third pulses. While
indeed producing an adiabatic refocusing pulse
the power required for this pulse is prohibitive for
in vivo applications.
A second proposal along similar lines is to only
use the even echoes in a spin echo experiment
in which sech pulses are used for refocusing (52).
This has lower power than the previously mentioned composite pulse but represents a pulse
sequence rather than a single RF pulse. The
power deposition may be further reduced by
simultaneously modulating both the gradient and
the RF fields using the VERSE technique (53).
A second group of methods is based on using
the properties of BIR-4. The first of these is
termed GMAX (for gradient modulated adiabatic
excitation) (54). The principle here is to give the
magnetic field gradient the same temporal modulation as the frequency offset of an adiabatic excitation pulse. There will be a node in this pulse
RF PULSE FORMS IN BIOMEDICAL NMR
Table 2
99
BISS-8 Pulse Scheme for the B1 t t, and Gt Fieldsa
Period
B1 t
t
Gt
1
2
3
4
5
6
7
8
+⇒0
0⇒+
0⇒+
0⇒+
−⇒0
−⇒0
+⇒0
0⇒−
0⇒+
0⇒+
+⇒0
−⇒0
+⇒0
0⇒−
0⇒+
0⇒+
+⇒0
−⇒0
+⇒0
0⇒+
0⇒+
0⇒+
−⇒0
−⇒0
a
The notation shows the direction of the modulation for the field in question; for example, − ⇒ 0 denotes a value which is
initially negative and is brought to zero.
at the point where the maximum offset induced
by the gradient field is cancelled exactly by the
frequency offset of the pulse: this effect is the
same as that discussed previously for the use of
the half-secant pulse, which is shown in Fig. 6.
This node is the center of a relatively sharp transition region across which the polarity of the transverse magnetisation is reversed. The excitation is
then performed a second time with the polarity
of the B1 -field reversed and the position of the
node shifted by modifying the gradient strength
and/or the amplitude of frequency modulation of
the pulse. The positions of the two nodes delimit
the extent of the selected slice obtained after addition of the two signals. Unfortunately this process
is a two step experiment which makes it vulnerable to motion and other instabilities; furthermore the magnetization outside the excited slice
is perturbed and multislice experiments are hence
impractical.
In the BISS-8 experiment (55) two BIR-4 pulses
are combined, with one of the pulses sandwiched
between the two halves of the other, as shown in
Table 2. The total excitation angle () is given by
the sum of the phase discontinuities as follows:
1 = −4 = 180◦ +
4
2 = −3 = −180 +
4
[33]
◦
The gradient modulation function during the
third to sixth periods (i.e., the central pulse of
the two) is the inverse of the frequency modulation function, and so in regions where the gradient field dominates the frequency modulation the
two pulses produce equal and opposite excitations.
The excitation angle outside the region where the
frequency offset of the pulse is greater than that
induced by the gradient field is hence zero. It
was subsequently noted that a slice selective excitation could be generated using a single BIR-4
pulse 56 57. In the mASSESS pulse of Shen
and Rothman (56) and the SLAB-4 of Hsu et al.
(57), a BIR-4 pulse is played out in which during
the first half of the pulse the gradient and frequency modulations have the same polarity, and
in the second half of the pulse they have opposite polarity: in the regions where the frequency
offset always dominates the magnetization experiences a standard BIR-4 pulse, whereas in those
regions where the gradient induced offset dominates for one half of the pulse the excitation angle
will be zero. This scheme is given in Table 3. The
main practical problem encountered with all these
pulses is the high slew-rate requirement for the
magnetic field gradients, and the need to instantaneously switch the polarity of the gradients midway through the pulse. The ASSESS pulse of Shen
and Rothman (56) has a smoothly varying gradient modulation function of constant polarity: this
means that during the first and third sections of
BIR-4 the amplitude and frequency modulations
are in antiphase. This pulse generates a satisfactory in-plane excitation and an out of plane saturation, which makes it potentially of interest for
spectroscopy applications but unsuitable for multislice imaging. The scheme of this pulse is given
in Table 4.
In conclusion, the lack of a general formalism
for generating adiabatic pulses means that slice
selective pulses have only been generated by vectorial arguments. None of the pulses so far proposed is ideal in its performance, the gradient
requirements in particular being taxing for in vivo
applications especially on whole-body systems.
Table 3 Pulse Scheme for the mASSESS/SLAB-4
Scheme Using the Same Notation as in Table 2
Period
B1 t
t
Gt
1
2
3
4
+⇒0
0⇒+
0⇒+
0⇒+
−⇒0
−⇒0
+⇒0
0⇒+
0⇒−
0⇒+
−⇒0
+⇒0
100
NORRIS
Table 4 Pulse Scheme for the ASSESS Scheme
Using the Same Notation as in Table 2
Period
B1 t
t
Gt
1
2
3
4
+⇒0
0⇒−
0⇒+
0⇒+
+⇒0
+⇒0
+⇒0
0⇒−
0⇒+
0⇒+
+⇒0
+⇒0
ACKNOWLEDGMENT
The author thanks Thies Jochimsen for assistance
in producing diagrams 3 and 4.
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David G. Norris received his first degree
in Physics from Cambridge University and
subsequently obtained an M.Sc. and a
Ph.D. from the University of Aberdeen.
His doctoral thesis examined the subject
of NMR flow imaging. He subsequently
worked for eight years at the University of Bremen concentrating primarily on
the themes of fast imaging and diffusion,
before taking an appointment as head of the NMR group at
the newly founded Max-Planck-Institute for Cognitive Neuroscience in Leipzig. Here he developed further interests in
functional MRI and perfusion imaging. He is now a Principal
Investigator and head of NMR at the FC Donders Centre for
Cognitive Neuroimaging in Nijmegen.